This is Discrete Math Assignment on Recursive definitions and structural induction.

ecursively, mathematical induction is used very often, as the recursive definitions enable the use of mathematical nduction very naturally. Simple questions about recursively defined functions: 1. Find f (1), f (2), f(3), f (4) and f(5) if f (n) is defined recursively by f (0) = 1 and for n = 1, 2, 3, .. a. f ( n + 1) = f(n)+2. b. f (n + 1) = 3f(n). c. f (n + 1) = 25(n) d. f (n + 1) = f(n)2+ f(n)+1. 2. Determine whether each of these proposed definitions is a valid recursive definition of a function f from the of nonnegative integers to the set of integers. If f is well-defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid. a. f(0) = 0, f (n) = 2f(n - 2) forn 2 1. b. f(0) = 1, f(n) = f(n- 1) - 1 forn 2 1. c. f(0) = 2, f(1) = 3, f(n) = f(n - 1) - 1 for n 2 2. d. f(0) = 1, f(1) = 2, f (n) = 2f(n - 2) forn 2 2. e. f(0) = 1, f(n) = 3f(n - 1) if n is odd and n 2 1 and f(n) = 9f(n - 2) if n is even and n 2 2